Mathematical reformulation of the Kolmogorov–Richardson energy cascade in terms of vortex stretching

Abstract

In this paper, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov–Richardson energy cascade in terms of vortex stretching. By using the description, we prove that if the Navier–Stokes flow satisfies a new regularity criterion in terms of the enstrophy production rate, then the flow does not blow up. Our DNS results seem to support this regularity criterion. Next, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of the scale-locally of the vortex stretching/compressing (i.e. energy cascade) process and the statistical independence between vortices that are not directly stretched or compressed, we can derive the −5/3 power law of the energy spectrum of statistically stationary turbulence without directly using the Kolmogorov hypotheses.

Self-similar hierarchy of coherent tubular vortices in turbulence

“…From the hierarchy of coherent vortices obtained in this way, we provide a quantitative verification of the assumptions in Ref. [20]. We emphasize that this gives not only a verification of the mathematical assumptions but also physically meaningful knowledge.…”

“…These assumptions are physically reasonable and the scale-locality of the interaction was numerically verified in Ref. [20]. However, there is no quantitative verification of the assumptions on the decomposition itself.…”

“…Although such a process that the larger-scale vortex stretches and creates smaller-scale ones has long been proposed [4], recent DNS of developed turbulence at high Reynolds numbers make it possible to show that developed turbulence is indeed composed of coherent structures at various length scales. Furthermore, such DNS can capture concrete energy-cascading events [15,19] and quantify the scale-locality of energy cascade due to vortex stretching [16,20].…”

“…In Ref. [20], using this concrete picture [16] of energy cascade in terms of the hierarchy of coherent vortices, we proposed a new regularity criterion for a solution of the Navier-Stokes equation, and reformulated the energy cascade in developed turbulence. In particular, we derived the −5/3 power law of the energy spectrum from the Navier-Stokes equation without directly using the Kolmogorov similarity hypothesis [21].…”

“…In Ref. [20], we decomposed turbulence in a periodic cube into scales by using the band-pass filter and imposed conditions on the interaction between the scales in the stretching process. One of the most important assumptions on the interaction between scales is that the vorticity at each scale is expressed by coherent vortices (see (13) in §4(c) for the concrete expression).…”